| Gli Algoritmi della Libreria SOFA | ||
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Per prima cosa vediamo in dettaglio le applicazioni che Patrick Wallace ha riportato, da p.17 a p.27, nel suo ′cookbook′, ovvero il manuale-guida delle routine SOFA, intitolato “Sofa Tools for Earth Attitude”. Possiamo tradurlo come 'Strumenti SOFA per determinare la posizione della Terra nello spazio', in quanto il termine “attitude” sta per `direzione`.
Cominciamo con l'Example 5.2 di p.20-21 del documento suddetto. Riguarda la matrice di precessione-nutazione-bias NPB (equinox based) calcolata tenendo in considerazione le vecchie risoluzioni IAU 1976/1980/1982/1994. Il codice Fortran 90 del file 1.f90, ricco di informazioni riguardanti le impostazioni iniziali del problema e le varie matrici calcolate, è prelevabile da qui, mentre per l'eseguibile DOS (se non si ha tempo per compilarlo) basta scaricare, e poi dezippare, questo file.
L'output si ottiene con il comando: 1 > risu.txt ed è riportato appresso. Si noterà pure come nelle tre matrici sono rispettate le approssimazioni fino alla 15a decimale, ovvero sino alla doppia precisione delle variabili, assunta con il comando REAL(8) di Fortran 90. Naturalmente le stesse coincidono perfettamente con quelle di p.21 del “cookbook”.
(1.f90) p.20-21
IAU 1976/1980/1982/1994, equinox based
======================================
Greenwich Sideral Time (GST)= 13.412402380741 deg
0h 53m 38.976571378s
NPB MATRIX, equinox based
RNPB (3x3) = 0.999998403176203 -0.001639032970562 -0.000712190961847
0.001639000942243 0.999998655799521 -0.000045552846624
0.000712264667137 0.000044385492226 0.999999745354454
Celestial to Terrestrial MATRIX: no polar motion
RC2TI (3x3)= 0.973104317592263 0.230363826166889 -0.000703332813776
-0.230363798723540 0.973104570754695 0.000120888299841
0.000712264667137 0.000044385492226 0.999999745354454
Celestial to Terrestrial MATRIX: including polar motion
RC2IT (3x3)= 0.973104317712771 0.230363826174788 -0.000703163477127
-0.230363800391874 0.973104570648020 0.000118545116892
0.000711560100206 0.000046626645796 0.999999745754057
Proseguiamo adesso con gli altri tre esempi del manuale, inseriti insieme al primo esempio, nei seguenti listati:
E questi, in sequenza, sono i risultati:
(2.f90) p.22-23
IAU 2000A, CIO based, using classical angles
============================================
Earth Rotation Angle (ERA)= 13.318492966096 deg
0h 53m 16.438311863s
X=0.000712264729708; Y=0.000044385250265; s=-0.002200496 arcsec
NPB MATRIX, CIO based
RC2I (3x3) = 0.999999746339445 -0.000000005138721 -0.000712264730182
-0.000000026475329 0.999999999014975 -0.000044385242666
0.000712264729708 0.000044385250265 0.999999745354420
Celestial to Terrestrial MATRIX: no polar motion
RC2TI (3x3)= 0.973104317573104 0.230363826247805 -0.000703332818915
-0.230363798804278 0.973104570735551 0.000120888549767
0.000712264729708 0.000044385250265 0.999999745354420
Celestial to Terrestrial MATRIX: including polar motion
RC2IT (3x3)= 0.973104317697512 0.230363826239225 -0.000703163482268
-0.230363800456133 0.973104570632778 0.000118545366806
0.000711560162777 0.000046626403835 0.999999745754024
*-----*
(3.f90) p.23-24
2000A, equinox based
====================
Date: 2007- 4- 5 Time: 12: 0: 0.00 UT1-UTC=-0.07207 s (from IERS)
TT [MJD]: 54195.500754444445 = UTC corrected for leap seconds
UT1 [MJD]: 0.499999165814 = UTC + DeltaT (from IERS)
DATE [MJD]: 54195.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.500000000000 = UTC
--------------------------------
Greenwich Sideral Time (GST)= 13.412417084674 deg
0h 53m 38.980100322s
NPB MATRIX, equinox based
RNPB (3x3) = 0.999998402755640 -0.001639289519579 -0.000712191013215
0.001639257491365 0.999998655379006 -0.000045552787477
0.000712264729795 0.000044385250265 0.999999745354420
Celestial to Terrestrial MATRIX: no polar motion
RC2TI (3x3)= 0.973104317573210 0.230363826247358 -0.000703332818999
-0.230363798803831 0.973104570735657 0.000120888549787
0.000712264729795 0.000044385250265 0.999999745354420
Celestial to Terrestrial MATRIX: including polar motion
RC2IT (3x3)= 0.973104317697618 0.230363826238777 -0.000703163482352
-0.230363800455686 0.973104570632884 0.000118545366826
0.000711560162864 0.000046626403835 0.999999745754024
*-----*
(4.f90) p.26-27
IAU2006/2000A, CIO based, using X,Y series
==========================================
Date: 2007- 4- 5 Time: 12: 0: 0.00 UT1-UTC=-0.07207 s (from IERS)
TT [MJD]: 54195.500754444445 = UTC corrected for leap seconds
UT1 [MJD]: 0.499999165814 = UTC + DeltaT (from IERS)
DATE [MJD]: 54195.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.500000000000 = UTC
--------------------------------
Earth Rotation Angle (ERA)= 13.318492966096 deg
0h 53m 16.438311863s
X=0.000712264729525; Y=0.000044385248875; s=-0.002200475 arcsec
NPB MATRIX, CIO based
RC2I (3x3) = 0.999999746339445 -0.000000005138822 -0.000712264729999
-0.000000026475226 0.999999999014975 -0.000044385241276
0.000712264729525 0.000044385248875 0.999999745354420
Celestial to Terrestrial MATRIX: no polar motion
RC2TI (3x3)= 0.973104317573128 0.230363826247706 -0.000703332818416
-0.230363798804178 0.973104570735574 0.000120888551078
0.000712264729525 0.000044385248875 0.999999745354420
Celestial to Terrestrial MATRIX: including polar motion
RC2IT (3x3)= 0.973104317697536 0.230363826239126 -0.000703163481769
-0.230363800456033 0.973104570632801 0.000118545368117
0.000711560162594 0.000046626402444 0.999999745754025
A seguire, altri esempi. Eccoli (Aggiornamento del 17.11.2007).
Ecco pure i tabulati dei risultati degli ultimi 6 esempi.
(3set.f90)
IAU2006/2000A, CIO based, using X,Y series [Section A.3 test]
=============================================================
Date: 2006- 1-15 Time: 21:24:37.50 UT1-UTC= 0.3341 s
TT [MJD]: 53750.892855138889 = UTC corrected for leap seconds
UT1 [MJD]: 0.892104561343 = UTC + DeltaT (from IERS)
DATE [MJD]: 53750.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.892100694444 = UTC
--------------------------------
Earth Rotation Angle (ERA)= 76.265431053521 deg
5h 5m 3.703452845s
X(rad)=0.000584859819249; Y(rad)=0.000041535242468; s(”)=-0.002571986
X(”) = 120.635997299064; Y(”)= 8.567258740044
NPB MATRIX, CIO based --> (A.32)
RC2I = 0.99999982896948092 0.00000000032319160 -0.00058485981976671
-0.00000002461548598 0.99999999913741178 -0.00004153523517497
0.00058485981924879 0.00004153524246778 0.99999982810689292
Celestial to Terrestrial matrix --> (A.33)
RC2TI = 0.23742421473054331 0.97140604802742336 -0.00017920749985662
-0.97140588849284648 0.23742427873022337 0.00055827489333994
0.00058485981924879 0.00004153524246778 0.99999982810689292
*-----*
(7set.f90)
IAU2006/2000A, CIO based, using X,Y series [Section A.3 test]
=============================================================
Date: 2006- 1-15 Time: 21:24:37.50 UT1-UTC= 0.3341 s
TT [MJD]: 53750.892855138889 = UTC corrected for leap seconds
UT1 [MJD]: 0.892104561343 = UTC + DeltaT (from IERS)
DATE [MJD]: 53750.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.892100694444 = UTC
--------------------------------
Earth Rotation Angle (ERA)= 76.265431053521 deg
5h 5m 3.703452845s
X(rad)=0.000584859819249; Y(rad)=0.000041535242468; s(”)=-0.002571986
X,Y,Z= 0.00058485981924879 0.00004153524246778 0.99999982810689292rad
s,ERA= -0.00000001246933987 1.33108287733667652 rad
bet,a= 1.33108286486733673 0.50000004297328049 rad
r11,r12,r13= 0.23742426255380722 -0.97140604787395635 -0.00017920751377925
r21,r22,r23= -0.97140587680421864 0.23742427935812385 0.00055827488887075
Celestial to Terrestrial matrix --> (A.33)
RC2TI = 0.23742421473054331 0.97140604802742336 -0.00017920749985662
-0.97140588849284648 0.23742427873022337 0.00055827489333994
0.00058485981924879 0.00004153524246778 0.99999982810689292
*-----*
(8set.f90)
IAU2006/2000A, CIO based, using X,Y series [Section A.3 test]
=============================================================
Date: 2006- 1-15 Time: 21:24:37.50 UT1-UTC= 0.3341 s
TT [MJD]: 53750.892855138889 = UTC corrected for leap seconds
UT1 [MJD]: 0.892104561343 = UTC + DeltaT (from IERS)
DATE [MJD]: 53750.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.892100694444 = UTC
--------------------------------
Earth Rotation Angle (ERA)= 76.265431053521 deg
5h 5m 3.703452845s
X(rad)=0.000584859819249; Y(rad)=0.000041535242468; s(”)=-0.002571986
X,Y,Z= 0.00058485981924879 0.00004153524246778 0.99999982810689292 rad
s,ERA= -0.00000001246933987 1.33108287733667652 rad
E,d= 0.07089840047176134 0.00058633286103962 rad
ang1 = theta-E-s= 1.26018448933425509 rad
Celestial to Terrestrial matrix --> (A.33)
R = 0.23742421473054331 0.97140604802742336 -0.00017920749985662
-0.97140588849284648 0.23742427873022337 0.00055827489333994
0.00058485981924879 0.00004153524246778 0.99999982810689292
Matrix R, compact formulation --> R = R3(theta-E-s)*R2(d)*R3(E)
being: theta=ERA
E=atan2(X,Y)
d= atan(sqrt(X^2+Y^2/Z^2))
R = 0.23742421473054320 0.97140604802742336 -0.00017920749985662
-0.97140588849284637 0.23742427873022329 0.00055827489333994
0.00058485981924879 0.00004153524246778 0.99999982810689292
*-----*
(denver.f90)
IAU2006/2000A, CIO based, using X,Y series
==========================================
Date: 2004- 5-31 Time: 22: 0: 0.00 UT1-UTC=-0.47093 s (from IERS)
TT [MJD]: 53156.917409537033 = UTC corrected for leap seconds
UT1 [MJD]: 0.916661216088 = UTC + DeltaT (from IERS)
DATE [MJD]: 53156.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.916666666667 = UTC
--------------------------------
Given RA -> 23:32:55.171 Given Decl -> 52 16 38.29
Earth Rotation Angle (ERA)= 219.676330978484 deg
14h 38m 42.319s
NPB MATRIX, CIO based
RC2I(3x3) = 0.99999991755346973 0.00000000726944705 -0.00040607025673205
-0.00000002029049499 0.99999999948588580 -0.00003206599362056
0.00040607025629018 0.00003206599921620 0.99999991703935587
P0 -> initial geocentric versor
(x0,y0,z0)= 0.60757413831918361 -0.07212750078667660 0.79098121979979363
P1 -> final geocentric versor
(x1,y1,z1)= 0.60725289375548264 -0.07215287667632292 0.79122555912516102
Final RA (= RA1) -> 23h 32m 53.753s
Final DE (= DE1) -> 52° 18′ 0.683”
*-----*
(exa1.f90)
IAU2006/2000A, CIO based, using X,Y series [exa1.for]
=====================================================
Date: 2004- 5-31 Time: 22: 0: 0.00 UT1-UTC=-0.4709 s
TT [MJD]: 53156.917409537033 = UTC corrected for leap seconds
UT1 [MJD]: 0.916661216088 = UTC + DeltaT (from IERS)
DATE [MJD]: 53156.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.916666666667 = UTC
--------------------------------
Earth Rotation Angle (ERA)= 219.676330978484 deg
14h 38m 42.319434836s
X(rad)=0.000406070256290; Y(rad)=0.000032065999216; s(”)=-0.002842323
Lon(east)= -104.995000 deg; Lat= 39.742700 deg
alpha (RA)= 353.229879165 deg
23h 32m 55.171000s
Local Hour Angle (HA)= 121.451451813484 deg
8h 5m 48.348435s
*-----*
(exa2.f90)
IAU2006/2000A, CIO based, using X,Y series [exa1.for]
=====================================================
Date: 2004- 5-31 Time: 22: 0: 0.00 UT1-UTC=-0.4709 s
TT [MJD]: 53156.917409537033 = UTC corrected for leap seconds
UT1 [MJD]: 0.916661216088 = UTC + DeltaT (from IERS)
DATE [MJD]: 53156.000000000000 = UTC Date at 0 hrs
Fraction of day: 0.916666666667 = UTC
--------------------------------
Greenwich Sideral Time (GST)= 219.729877844908 deg
14h 38m 55.170682778s
Earth Rotation Angle (ERA)= 219.729877844908 deg
14h 38m 42.319434836s
NPB MATRIX, equinox based
RNPB (3x3) = 0.999999480862442 -0.000934561717244 -0.000406040936683
0.000934548620079 0.999999562783026 -0.000032444310370
0.000406071080366 0.000032064828530 0.999999917039059
Gamma Vector -> (13) of A&A 459 -> NPB 1st row
= 0.999999480862442 -0.000934561717244 -0.000406040936683
y Vector -> (14) of A&A 459 -> NPB 2nd row
= 0.000934548620079 0.999999562783026 -0.000032444310370
Sigma Vector -> (15) of A&A 459 -> NPB 3rd row
= 0.999999917553470 -0.000000006510525 -0.000406070256290
Equation of Origins EO = ERA-GST = -192.768719119 arcsec
= -12.851247941 s
... fine ...